Bifurcation analysis of an additional food-provided predator-prey system with anti-predator behavior

Year 2025, Volume: 5 Issue: 1, 38 - 64, 31.03.2025

Manoj Kumar Singh , Poonam Poonam

https://doi.org/10.53391/mmnsa.1496827

Abstract

This article analyzes the qualitative behavior of a predator-prey system where the predator receives extra food and the prey engages in anti-predator behavior to defend itself against attacks by the predator. The positivity and the boundedness of solutions to the system have been examined. The biologically well-posed equilibrium points of the proposed system are derived, and an analysis of their local stability is conducted. In specific situations, it is observed that the solutions of the proposed system are significantly dependent on the initial values. The emergence of several bifurcations in the system, including the saddle-node, Bogdanov-Takens, and Hopf-Andronov, is also shown. Through numerical simulation, the rise of a homoclinic loop is shown. The analytic results are verified by numerical simulations and phase portrait sketches.

Keywords

Additional food , anti-predator behavior , stability , bifurcation

References

• [1] Strauss, S.Y. Indirect effects in community ecology: their definition, study and importance. Trends in Ecology & Evolution, 6(7), 206-210, (1991).
• [2] Savitri, D. Dynamics analysis of anti-predator model on intermediate predator with ratio dependent functional responses. In Proceedings, The 2nd International Joint Conference on Science and Technology (IJCST), pp. 012201-012206, Bali, Indonesia, (2017, September).
• [3] Murdoch, W.W., Chesson, J. and Chesson, P.L. Biological control in theory and practice. The American Naturalist, 125(3), 344-366, (1985).
• [4] Haque, M. and Greenhalgh, D. When a predator avoids infected prey: a model-based theoretical study. Mathematical Medicine and Biology, 27(1), 75-94, (2010).
• [5] Holt, R.D. and Lawton, J.H. The ecological consequences of shared natural enemies. Annual Review of Ecology, Evolution, and Systematics, 25, 495-520, (1994).
• [6] Holt, R.D. Predation, apparent competition, and the structure of prey communities. Theoretical Population Biology, 12(2), 197-229, (1977).
• [7] Sahoo, B. and Poria, S. Disease control in a food chain model supplying alternative food. Applied Mathematical Modelling, 37(8), 5653-5663, (2013).
• [8] Srinivasu, P.D.N., Prasad, B.S.R.V. and Venkatesulu, M. Biological control through provision of additional food to predators: a theoretical study. Theoretical Population Biology, 72(1), 111-120, (2007).
• [9] Wade, M.R., Zalucki, M.P., Wratten, S.D. and Robinson, K.A. Conservation biological control of arthropods using artificial food sprays: current status and future challenges. Biological Control, 45(2), 185-199, (2008).
• [10] Prasad, B.S.R.V., Banerjee, M. and Srinivasu, P.D.N. Dynamics of additional food provided predator–prey system with mutually interfering predators. Mathematical Biosciences, 246(1), 176-190, (2013).
• [11] Sahoo, B. and Poria, S. Effects of supplying alternative food in a predator–prey model with harvesting. Applied Mathematics and Computation, 234, 150-166, (2014).
• [12] Chakraborty, K. and Das, S.S. Biological conservation of a prey-predator system incorporating constant prey refuge through provision of alternative food to predators: a theoretical study. Acta Biotheoretica, 62, 183-205, (2014).
• [13] Sen, M., Srinivasu, P.D.N. and Banerjee, M. Global dynamics of an additional food provided predator-prey system with constant harvest in predators. Applied Mathematics and Computation, 250, 193-211, (2015).
• [14] Shome, P., Maiti, A. and Poria, S. Effects of intraspecific competition of prey in the dynamics of a food chain model. Modeling Earth Systems and Environment, 2, 1-11, (2016).
• [15] Ghosh, J., Sahoo, B. and Poria, S. Prey-predator dynamics with prey refuge providing additional food to predator. Chaos, Solitons & Fractals, 96, 110-119, (2017).
• [16] Singh, M.K. and Bhadauria, B.S. Qualitative analysis of an additional food provided predator–prey model in the presence of Allee effect. International Journal of Applied and Computational Mathematics, 3(Suppl 1), 1173-1195, (2017).
• [17] Das, A. and Samanta, G.P. A prey–predator model with refuge for prey and additional food for predator in a fluctuating environment. Physica A: Statistical Mechanics and its Applications, 538, 122844, (2020).
• [18] Thirthar, A.A., Majeed, S.J., Alqudah, M.A., Panja, P. and Abdeljawad, T. Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator. Chaos, Solitons & Fractals, 159, 112091, (2022).
• [19] Debnath, S., Majumdar, P., Sarkar, S. and Ghosh, U. Memory effect on prey–predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey. Journal of Computational Science, 66, 101929, (2023).
• [20] Ananth, V.S. and Vamsi, D.K.K. Time optimal control studies and sensitivity analysis of additional food provided prey-predator systems involving Holling type III functional response based on quality of additional food. Journal of Biological Systems, 31(01), 271-308, (2023).
• [21] Das, B.K., Sahoo, D. and Samanta, G. Fear and its carry-over effects in a delay-induced predator-prey model with additional food to predator. Filomat, 37(18), 6059-6088, (2023).
• [22] Umaroh, S.Z. and Savitri, D. Dynamic analysis of a prey predator model with Holling-type III functional response and anti-predator behavior. Jurnal Sains, Teknologi dan Industri, 21(1), 51-57, (2023).
• [23] Berryman, A.A. The origins and evolution of predator-prey theory. Ecology, 73(5), 1530-1535, (1992).
• [24] Ford, J.K. and Reeves, R.R. Fight or flight: antipredator strategies of baleen whales. Mammal Review, 38(1), 50-86, (2008).
• [25] Ge, D., Chesters, D., Gomez-Zurita, J., Zhang, L., Yang, X. and Vogler, A.P. Anti-predator defence drives parallel morphological evolution in flea beetles. Proceedings of the Royal Society B: Biological Sciences, 278(1715), 2133-2141, (2011).
• [26] Lima, S.L. Nonlethal effects in the ecology of predator-prey interactions. Bioscience, 48(1), 25-34, (1998).
• [27] Matassa, C.M., Donelan, S.C., Luttbeg, B. and Trussell, G.C. Resource levels and prey state influence antipredator behavior and the strength of nonconsumptive predator effects. Oikos, 125(10), 1478-1488, (2016).
• [28] Zanette, L.Y., White, A.F., Allen, M.C. and Clinchy, M. Perceived predation risk reduces the number of offspring songbirds produce per year. Science, 334(6061), 1398-1401, (2011).
• [29] Panja, P., Mondal, S.K. and Chattopadyay, J. Dynamical effects of anti-predator behavior of adult prey in a predator-prey model with ratio-dependent functional response. Asian Journal of Mathematics and Physics, 1(1), 19-32, (2017).
• [30] Khater, M., Murariu, D. and Gras, R. Predation risk tradeoffs in prey: effects on energy and behaviour. Theoretical Ecology, 9, 251-268, (2016).
• [31] Samanta, S., Mandal, A.K., Kundu, K. and Chattopadhyay, J. Control of disease in prey population by supplying alternative food to predator. Journal of Biological Systems, 22(04), 677-690, (2014).
• [32] Tang, B. and Xiao, Y. Bifurcation analysis of a predator-prey model with anti-predator behaviour. Chaos, Solitons & Fractals, 70, 58-68, (2015).
• [33] Mortoja, S.G., Panja, P. and Mondal, S.K. Dynamics of a predator-prey model with stagestructure on both species and anti-predator behavior. Informatics in Medicine Unlocked, 10, 50-57, (2018).
• [34] Prasad, K.D. and Prasad, B.S.R.V. Qualitative analysis of additional food provided predator-prey system with anti-predator behaviour in prey. Nonlinear Dynamics, 96, 1765-1793, (2019).
• [35] Sahoo, B., Das, B. and Samanta, S. Dynamics of harvested-predator–prey model: role of alternative resources. Modeling Earth Systems and Environment, 2, 140, (2016).
• [36] Chen, J., Huang, J., Ruan, S. and Wang, J. Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting. SIAM Journal on Applied Mathematics, 73(5), 1876-1905, (2013).
• [37] Huang, J., Gong, Y. and Chen, J. Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting. International Journal of Bifurcation and Chaos, 23(10), 1350164, (2013).
• [38] Perko, L. Differential Equations and Dynamical Systems (Vol. 7). Springer: New York, 2001.